Metamaterials are artificial materials that are constructed to have particular properties, typically properties that are not readily found in natural materials. An example of metamaterials is microstructured optical materials constructed to have a very low transmissivity of light over a range of electromagnetic frequencies. Those frequencies are typically referred to as a photonic bandgap, in analogy to the bandgaps seen in the behaviour of electrons in semiconductors. At least some photonic bandgaps can be understood in terms of Bragg scattering of photons in periodic microstructured material. The term “bandgap” is often used to describe a low transmissivity resulting from the structure of the metamaterials, even when a “complete” bandgap (as defined by Bloch-Floquet theory) is not observed.
Subsequent to the development of metamaterials exhibiting such (full or partial) photonic bandgaps, attempts were made to develop metamaterials exhibiting phononic bandgaps, i.e. regions in the frequency spectrum in which the transmissivity of sound quanta or phonons is greatly reduced, leading to very high levels of attenuation. Moreover, unlike traditional periodic materials that have been employed at high frequencies, acoustic metamaterials can include resonant elements that allow band gaps to form within the long wavelength limit. It is at low frequencies where it is most difficult to design satisfactory passive isolation solutions, and hence metamaterials may provide a useful path to high-performance, low-frequency isolation.
The behaviour of locally resonant photonic and phononic metamaterials can be understood in terms of their so-called “left-handedness”. In optical material having simultaneously a negative electromagnetic permeability and a negative electromagnetic permittivity at some frequencies (when the material is said to be in its double negative or DNG region), the refractive index has a negative sign so negative refraction occurs. Snell's law of refraction still applies but, because the refractive index is negative, the path of the reflected wave lies on the opposite side of the normal to the reflecting surface to that one would expect in a transmission medium having a positive refractive index, and the group and phase velocity vectors are anti-parallel.
In acoustic metamaterials, the analogues of permeability and permittivity are density and bulk modulus, respectively. Thus, negative refraction occurs when the density and bulk modulus of the metamaterials are simultaneously negative. As discussed above, and as with optical metamaterials, gaps appear in the dispersion characteristics of metamaterials, at high frequencies due to Bragg scattering effects related to the periodic properties of the metamaterial. In metamaterials where low frequency resonances occur, gaps can also occur at lower, resonant frequencies, typically at frequencies around two orders of magnitude lower than the Bragg bandgaps. This leads to high levels of attenuation in the transmission characteristics of the material at these frequencies. Such low sound transmissivity has the potential to produce novel acoustic behaviour beyond that seen in naturally occurring media, with potential applications including acoustic cloaking, transmission blocking, and sub-wavelength acoustic lenses.
An example (FIG. 1(a)) of an acoustic metamaterial having a frequency region with a negative bulk modulus is formed from an array of Helmholtz resonators 10 connected to a one-dimensional fluid transmission medium 20.
Other arrangements have been shown to possess either a negative effective modulus of elasticity or density. In the single negative (SNG) band those systems thus have a complex refractive index which acts to partially block the propagation of a pressure wave through the medium. Pope, S. A. and Daley, S. (2010) “Viscoelastic locally resonant double negative metamaterials with controllable effective density and elasticity”. Physics Letters A, 374, 4250-4255 model (FIG. 1b) the Helmholtz resonator 10 and fluid transmission region 20 as a sequence of identical pairs of cells, each pair being formed by a Helmholtz resonator element 10 and a fluid transmission element 20′. Each Helmholtz resonator element 10 is modelled as a mass mr having displacement xrr connected to a stationary reference point 30 through a stiffness element ch, kh and to the transmission medium 20 through a damping element cr, kr. The transmission medium 20′ itself is modelled as a mass m, to which the damping element cr, kr is connected, itself connected to adjacent masses m by stiffness elements k, c. Effective material parameters are derived using the classical laws of motion in the form of D'Alembert's principle, assuming linear viscoelastic material properties. It is shown that a cascade of Helmholtz resonators modelled in that way, i.e. such that they are dynamically equivalent to a mass with two separate elastic connections, one to the transmission medium and the other to a stationary reference, constitutes a medium capable of providing a negative effective density but not a negative effective bulk modulus.
Pope and Daley also show that an array of Helmholtz resonators can be modelled (FIG. 1(c)) by an array of locally controlled masses mr with a single elastic connection through the damping element cr, kr to the transmission medium 20. Although an array of masses mr with a single elastic connection to the transmission medium 20 again only provides a system with negative effective mass, not a negative bulk modulus, a local active control scheme fcn applied to each of the masses mr can emulate additional elastic connections to the supporting structure. An array of masses with a suitable local control scheme can provide both the negative effective stiffness and mass required for negative refraction. The tuneable feedback control parameters determine the characteristics of the region of double negativity. Thus, by extending the control system to take into account the motion of the transmission network, an effective homogeneous medium with the possibility of both a negative effective density and elasticity can be realised.
However, active control of the masses mr is less desirable than passive control, as it is more complex. Also, it would be desirable to extend the range of frequencies over which a negative refractive index is achievable. Furthermore, Pope and Daley's prior-art system is a theoretical proposal, and it would be desirable to provide a real-world structure exhibiting significantly reduced acoustic transmissivity, relative to comparable prior-art materials.